# Gr.group Theory – Group with character levels {1, pq, pr, qr}, where p, q and r are distinct prime numbers

I'm currently trying to link the derived length of some soluble groups by assuming that they possess only two degrees of irreducible monomial complex character. Using induction, it is often enough to consider the groups $$G$$ satisfactory that its set of irreducible degrees of character, $$textrm {cd} (G)$$satisfied

$$textrm {cd} (G) = lbrace 1, pq, pr, qr rbrace$$

or $$p, q$$ and $$r$$ are distinct prime numbers. In particular, I would like to know if these (resolvable) groups have a calculated length at most equal to 3, but I have not been able to prove it – also with the additional assumption that groups have only two degrees of monomial character. The paper

M.L.Lewis. Determination of the structure of a group from irreducible sets of character levels – Journal of Algebra (1997)

studies similar groups, where Mark Lewis is able to prove strong statements about the groups he is studying. This might suggest that it is reasonable to think that one can also apply similar restrictions to the groups above (and in particular to link by 3 its derived length).